Properties of Gröbner bases under specializations

Abstract

In this paper we prove some properties of Gröbner bases under specialization maps. In particular we state sufficient conditions for the image of a Gröbner basis to be a Gröbner basis. We apply these results to the resolution of systems of polynomial equations. In particular we show that, if the system has a finite number of solutions, (in an algebraic closure of the base field K), the problem is totally reduced to a single Gröbner basis computation (w.r.t. purely lexicographical ordering), followed by a search for the roots of univariate polynomials and a “few” evaluations in suitable algebraic extensions of K.